A intuitive explanation for the chain rule

The chain allows us to take derivative of composition of two functions.  It has the form
$$(f\circ g)'(x)=f'(g(x))g'(x)$$

Intuitively, why should it be a product of the two function's individual derivatives?

We can answer that by looking at Taylor expansions.
If we wiggle $x$ in $g(x)$ we get
$$g(x+\delta)\approx g(x)+g'(x)\delta$$

That means if we wiggle $x$ by $\delta$, f gets wiggled by $g'(x)\delta$.

A function \f(y)\ would change in a very similar way if we wiggle y:
$$f(y+\delta') \approx f(y) + f'(y)\delta'$$

We replace $\delta'$ by $g'(x)\delta$ because that is how much the "y" is perturbed if we perturb $x$ by $\delta$

Overall, we would get
$$f(g(x+\delta)) \approx f(g(x))+f'(g(x))g'(x)\delta$$